CAT (QA) PYQs

Let A be the largest positive integer that divides all the numbers of the form $3^k + 4^k + 5^k$, and B be the largest positive integer that divides all the numbers of the form $4^k + 3(4^k) + 4^{k + 2}$, where k is any positive integer. Then (A + B) equals

For natural numbers $x$, $y$, and $z$, if $xy + yz = 19$ and $yz + xz = 51$, then the minimum possible value of $xyz$ is

Alex invested his savings in two parts. The simple interest earned on the first part at $15 \%$ per annum for $4$ years is the same as the simple interest earned on the second part at $12 \%$ per annum for $3$ years. Then, the percentage of his savings invested in the first part is

Let a and b be natural number. If $a^2 + ab + a = 14$ and $b^2 + ab + b = 28$, then $(2a + b)$ equals

In a class of $100$ students, $73$ like coffee, $80$ like tea and $52$ like lemonade. It may be possible that some students do not like any of these three drinks. Then the difference between the maximum and minimum possible number of students who like all the three drinks is

A mixture contains lemon juice and sugar syrup in equal proportion. If a new mixture is created by adding this mixture and sugar syrup in the ratio $1: 3$, then the ratio of lemon juice and sugar syrup in the new mixture is

Let $a, b, c$ be non-zero real numbers such that $b^2 < 4ac$, and $f(x) = ax^2 + bx + c$. If the set S consists of all integers m such that $f(m) < 0$, then the set S must necessarily be

Amal buys $110 \mathrm{~kg}$ of syrup and $120 \mathrm{~kg}$ of juice, syrup being 20\% less costly than juice, per kg. He sells $10 \mathrm{~kg}$ of syrup at 10\% profit and $20 \mathrm{~kg}$ of juice at $20 \%$ profit. Mixing the remaining juice and syrup, Amal sells the mixture at Rs. $308.32$ per kg and makes an overall profit of $64 \%$. Then, Amal's cost price for syrup, in rupees per kg, is

In a village, the ratio of number of males to females is $5: 4$. The ratio of number of literate males to literate females is $2 : 3$. The ratio of the number of illiterate males to illiterate females is $4: 3$. If $3600$ males in the village are literate, then the total number of females in the village is

All the vertices of a rectangle lie on a circle of radius $R$. If the perimeter of the rectangle is $P$, then the area of the rectangle is

Let $ABCD$ be a parallelogram such that the coordinates of its three vertices $A, B, C$ are $(1, 1), (3, 4)$ and $(-2, 8)$, respectively. Then, the coordinates of the vertex $D$ are

The largest real value of a for which the equation $|x + a| + |x - 1| = 2$ has an infinite number of solutions for $x$ is

The number of ways of distributing $20$ identical balloons among $4$ children such that each child gets some balloons but no child gets an odd number of balloons, is

For any natural number n, suppose the sum of the first n terms of an arithmetic progression is $(n + 2n^2)$. If the $n^{th}$ term of the progression is divisible by $9$, then the smallest possible value of $n$ is

A trapezium $ABCD$ has side $AD$ parallel to $BC$, $\angle$BAD = $90°$, $BC = 3$cm and $AD = 8$cm. If the perimeter of this trapezium is $36$cm, then its area, in sq. cm, is

Let $0 \leq a \leq x \leq 100$ and $f(x)=|x-a|+|x-100|+|x-a-50|$. Then the maximum value of $f(x)$ becomes $100$ when $a$ is equal to

The average of three integers is $13$. When a natural number $n$ is included, the average of these four integers remains an odd integer. The minimum possible value of $n$ is

For any real number x, let [x] be the largest integer less than or equal to x. If $\sum_{n=1}^{N} [\frac{1}{5} + \frac{n}{25}] = 25$ then N is

Ankita buys $4 \mathrm{~kg}$ cashews, $14 \mathrm{~kg}$ peanuts and $6 \mathrm{~kg}$ almonds when the cost of $7 \mathrm{~kg}$ cashews is the same as that of $30 \mathrm{~kg}$ peanuts or $9 \mathrm{~kg}$ almonds. She mixes all the three nuts and marks a price for the mixture in order to make a profit of Rs.1752. She sells $4$ kg of the mixture at this marked price and the remaining at a $20 \%$ discount on the marked price, thus making a total profit of Rs. 744. Then the amount, in rupees, that she had spent in buying almonds is

The average weight of students in a class increases by $600$ gm when some new students join the class. If the average weight of the new students is $3$ kg more than the average weight of the original students, then the ratio of the number of original students to the number of new students is

Pinky is standing in a queue at a ticket counter. Suppose the ratio of the number of persons standing ahead of Pinky to the number of persons standing behind her in the queue is $3: 5$. If the total number of persons in the queue is less than $300$, then the maximum possible number of persons standing ahead of Pinky is

Trains A and B start traveling at the same time towards each other with constant speeds from stations X and Y, respectively. Train A reaches station Y in $10$ minutes while train B takes $9$ minutes to reach station X after meeting train A. Then the total time taken, in minutes, by train $B$ to travel from station $Y$ to station $X$ is